Lee and Cruz have been the two Republicans in the Senate most vocal in the attempt to use a government shutdown as leverage to defund Obamacare. The two senators, who were among the Republicans that McCain derided as "wackobirds" earlier this year, are the most likely to use Senate rules to attempt to delay any legislation to avoid hitting the debt ceiling. Unless a bill is given "unanimous consent," it is required to be debated for 30 hours. Such a delay would put the government perilously close to default once the debt ceiling is breached on Thursday.

Sen. John Thune (R-SD) expressed his hope that such delaying tactics would not be used, although he admitted it's "not virtually impossible to know." However, Thune acknowledged that individual senators had the rights and prerogratives to throw up procedural hurdles.

A spokesperson for Senator Cruz said that the Texas senator had "a previous commitment that went long." He had been planning on attending the meeting of Senate Republicans, which was scheduled for 11:00 AM but was then unable to attend once it was rescheduled. Senator Lee's office has not responded to a request for comment.